Theorem eqglact 19059

Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	⊢ 𝑋 = (Base‘𝐺)
eqger.r	⊢ ∼ = (𝐺 ~QG 𝑌)
eqglact.3	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
eqglact	⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

Distinct variable groups:   𝑥, +   𝑥, ∼   𝑥,𝐺   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌

Proof of Theorem eqglact

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqger.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
2		eqid 2733	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
3		eqglact.3	. . . . . . 7 ⊢ + = (+g‘𝐺)
4		eqger.r	. . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑌)
5	1, 2, 3, 4	eqgval 19057	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
6		3anass 1096	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
7	5, 6	bitrdi 287	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌))))
8	7	baibd 541	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
9	8	3impa 1111	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
10	9	abbidv 2802	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∣ 𝐴 ∼ 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
11		dfec2 8706	. . 3 ⊢ (𝐴 ∈ 𝑋 → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
12	11	3ad2ant3 1136	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
13		eqid 2733	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥))) = (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))
14	13, 1, 3, 2	grplactcnv 18926	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴))))
15	14	simprd 497	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)))
16	13, 1	grplactfval 18924	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
17	16	adantl 483	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
18	17	cnveqd 5876	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
19	1, 2	grpinvcl 18872	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
20	13, 1	grplactfval 18924	. . . . . . . 8 ⊢ (((invg‘𝐺)‘𝐴) ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
21	19, 20	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
22	15, 18, 21	3eqtr3d 2781	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
23	22	cnveqd 5876	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
24	23	3adant2 1132	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
25	24	imaeq1d 6059	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌))
26		imacnvcnv 6206	. . 3 ⊢ (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)
27		eqid 2733	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥))
28	27	mptpreima 6238	. . . 4 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌}
29		df-rab 3434	. . . 4 ⊢ {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
30	28, 29	eqtri 2761	. . 3 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
31	25, 26, 30	3eqtr3g 2796	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
32	10, 12, 31	3eqtr4d 2783	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  {crab 3433   ⊆ wss 3949   class class class wbr 5149   ↦ cmpt 5232  ^◡ccnv 5676   “ cima 5680  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7409  [cec 8701  Basecbs 17144  +_gcplusg 17197  Grpcgrp 18819  inv_gcminusg 18820   ~_QG cqg 19002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-ec 8705  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-eqg 19005

This theorem is referenced by:  eqgen  19061  cldsubg  23615  tgpconncompeqg  23616  snclseqg  23620  pzriprnglem10  46814